All Bitcoin transactions are carried out by computers. There are no physical coins. It is somewhat like PayPal in this regard, but differs in several important ways, the most important being that bitcoins (BTC) are not dollars (although they can be exchanged); it is a separate currency entirely.
Bitcoin is not regulated by any central authority. Instead, the service relies on peer-to-peer connections between its users. Each and every transaction is announced and propagated throughout the network of Bitcoin users, who's clients store the information. Every ten minutes or so, this transaction information is formed into blocks by miners (we'll discuss that later) and is added to the block chain. This block chain serves as the master transaction record, and a copy of it is held by every user.
You may be wondering: "But, if everyone has a copy of every transaction ever, how can I retain my privacy?" The answer is this: Bitcoin does not use account numbers, it uses keys. Instead of sending BTC to an account, you send them to a randomly generated unique address. Each person can have as many addresses as they want, and any BTC sent to any of them goes into that person's wallet. To anyone looking at the transaction records, there is no connection between any of a person's addresses. This operates on the principle of public key cryptography; read here and here for more details.
Now, as promised, miners. Miners are Bitcoin users who create blocks for the block chain in exchange for BTC. For every block a miner creates, they receive some BTC; this is how bitcoins are "minted." There is a set limit to the total number of bitcoins that will ever exist; there will never be more than about 21 million. A user creates a block by dedicating their computer's CPU and GPU to finding a solution to a cryptographic problem. The first to find a solution gets the reward. The difficulty of the problem is adjusted regularly to make sure that only one block is made every 10 minutes.
Mining will most likely not be profitable for you, seeing as it could take several years for you to find a single block, and by then your electricity bill would have already exceeded your profits. It is a practice best left to those who know what they are doing.
Bitcoin is a fascinating currency system, and I hope that it continues to grow and thrive. Perhaps it will eventually be adopted by various vendors such as Amazon. If you want to learn more about Bitcoin (or if you are confused by my explanation), you can check out this site. If you feel like donating any BTC in my general direction, you can send them to:
Wow, that was a long post.
P.S. Uhh Yeah Dude is pretty funny.